Shizhong suggested a new way besides the eq. \eqref{eq:wfSemiCooper}, it is the normal two-body eqaution, except that the summation of the interaction is $\sum_{\vk'}{}U_{\vk'\vk}\psi_{\vk'}(1-n_{\vk'})$.  In the BEC side, $n_\vk\sim\nth{(a_sk_f)^3}$ and this should give me something. 
